Abstract:

The Hebrew (Jewish)
Calendar Paradox is, as far as I know, the only true logical paradox that
is the direct outgrowth of a legal system. It is caused by the fact that
in Jewish legal system, there is a law that defines time measurement and
time measurement is used to rule whether a law is valid or not. This gives
rise to circular reference and, more specifically, to what I call the vicious
Mobius reference.

The Paradox Definition
-- The Conditions that Set Forth the Paradox:

The Hebrew
Calendar Paradox is the result of a specific combination of Jewish laws:

The Hebrew
day starts at the evening and spans the night and the following daytime
till the next evening.

On any day, a judge can declare
that the current day is the first day of a Hebrew
month if at least two witnesses testify that they did not see the moon
during the previous night. (We can assume that the law presumed
that, if the moon were visible, the witnesses would have seen it; that
is, the skies were not obscured by clouds and the witnesses were in a position
and alert to see the moon throughout the night.)

It is important to keep in
mind the following fact: This legal discussion takes place during the day
time after the beginning of the day in question. In other words, the judge
has to determines weather or not an even took place during the previous
evening and, if it did, it signaled that the current day is the first day
of a new month. That is, the judge's decision is retroactive.

A single witness, who testifies
that he did see the moon, is sufficient to refute these two witnesses.
In this case the judge may not declare the day as the first of the month.

A person is a valid witness
if he is an adult man. [Note that masculinity has in fact nothing to do
with this paradox.]

An adult man is one who is at
least thirteen-year old.

Effects:

The effect of law #1 and law
#2, taken together, is that the beginning of a new month can be determined
only in retrospect. That is, if on a certain day a judge
declares that this day is the first day of a month, then the new month
started on previous evening.

Considering the following scenario
some ancient wise rabbis declared a stalemate (In Hebrew they said: "Tyku",
the acronym for "Tishbi Yitaretz Kooshiot Ve’ba’ayot", which roughly translates
as "the Messiah will explain questions and problems.")

One morning two witnesses
come to court and each of them declare to the judge that he did not see
the moon during the previous night. A boy comes forward and testifies that
he did see the moon. The judge asks the boy for his age and the boy replies:
"On the first of the month I will be thirteen."If the judge accepts the testimony
of the two witnesses, then the boy is thirteen-year old and thus his testimony
is valid. Therefore, he refutes the two witnesses. Rejecting their testimony
means that it is not the first day of the month and thus the boy is too
young to qualify as a witness. So he cannot refute the two witnesses and
the judge must accept their testimony that a new month has started. In
which case the boy is of age and...

In modern times it is clear
that root of the paradox is a circular dependency: A matter of law defines
a matter of time and a matter of time defines a matter of law. Evidently
the wise rabbis did not recognize that such reasoning is not valid.

The paradox cannot be avoided
by altering the number of witnesses or counter witnesses or by changing
the qualifying age to be a witness. For example, If one must be at least
thirteen-year old and one day or at least one day less than thirteen-year
old there may be a case in which a boy witness who is exactly a day older
or a day younger than thirteen on the first of the month. In other words,
this paradox is independent of the number of witnesses and counter witnesses
and the age at which a the witness' testimony becomes valid.

A negating twist must exist
amongst a finite set of proposition for such a paradox to occur. Indeed,
this set of propositions contains such a negation twist. The Mobius
vicious circle principle, as I call it, is a necessary and sufficient
condition to generate such paradoxes as Russell's paradox and the liar's
paradox.

Open Questions:

All of the information I have
about the Hebrew Calendar Paradox is based
on childhood recollections. I do not know for a fact that it truly exists.
Therefore, I am looking for any information that can confirm or refute
its existing, shed more light or provide any additional information about
it.